Analysis of Simulated Hungarian Income Data
Statistical Analysis Project
2025-05-17
Introduction
This analysis examines simulated income data for Hungary, focusing on the relationships between income and various demographic factors including age, location, occupation, and gender. The dataset is simulated to reflect realistic patterns while maintaining a manageable size for analysis.
Data Simulation
The data was simulated by the data_simulation.R script.
The data is available in the hungarian_income_data.csv
file.
Important things to note about the generated data:
Only the 8 most populated cities of Hungary are taken into count weighted by their population. List of cities: Budapest, Debrecen, Szeged, Miskolc, Pécs, Győr, Szombathely, Eger.
Only the 10 most common occupations are taken into count weighted by their frequency in the workforce. List of occupations: Software Developer, Teacher, Doctor, Sales Representative, Engineer, Accountant, Nurse, Manager, Chef, Driver.
The age distribution is generated by a beta distribution with parameters \(\alpha = 2\) and \(\beta = 3\) and multiplied by 95 to put the end result in the desired range. The beta distribution with the aforementioned parameters skews the age distribution towards younger ages, which is more realistic.
There are three groups of people categorized by their age:
Underage: each person has a random age at which they start working between 14 and 24.
Working age: 19-67
Pension age: each person has a random retirement age between 60 and 75.
Under 18 people have no income.
Working age people have a regular income based on their age, occupation, city, and gender.
Pension age people have a pension based on their occupation and city.
All working age people are considered to be employed.
The income of a working age man is 20.000 HUF higher than the income of a working age woman in the same occupation, city, and age group.
Demographics Analysis
library(forcats)
data <- data %>%
mutate(age_group = cut(age,
breaks = seq(0, 100, by = 2),
right = FALSE,
include.lowest = TRUE,
labels = seq(0, 98, by = 2)))
dem_pyramid <- data %>%
group_by(age_group, gender) %>%
summarise(count = n(), .groups = 'drop') %>%
mutate(count = ifelse(gender == "Male", -count, count))
ggplot(dem_pyramid, aes(x = age_group, y = count, fill = gender)) +
geom_bar(stat = "identity", width = 0.8, color = "black") +
scale_y_continuous(labels = abs, expand = expansion(mult = c(0.05, 0.05))) +
scale_fill_manual(values = c("Male" = "#00BFFF", "Female" = "#FF3B3B")) +
coord_flip() +
labs(title = "Population Pyramid of Simulated Hungarian Data",
x = "Age Group",
y = "Count",
fill = "Gender") +
custom_theme +
theme(legend.position = "top",
axis.text.y = element_text(size = 10, face = "bold"),
plot.margin = margin(t = 20, r = 20, b = 20, l = 20))
Data Preprocessing
detect_outliers <- function(x) {
q1 <- quantile(x, 0.25)
q3 <- quantile(x, 0.75)
iqr <- q3 - q1
lower_bound <- q1 - 1.5 * iqr
upper_bound <- q3 + 1.5 * iqr
return(x < lower_bound | x > upper_bound)
}
outliers <- detect_outliers(data$income)
data_clean <- data[!outliers, ]
set.seed(42)
income_age_matrix <- data_clean %>%
select(income, age) %>%
scale()
kmeans_result <- kmeans(income_age_matrix, centers = 3, nstart = 25)
data_clean$cluster <- kmeans_result$cluster
cluster_summary <- data_clean %>%
group_by(cluster) %>%
summarise(
mean_income = mean(income),
.groups = 'drop'
) %>%
arrange(mean_income)
cluster_labels <- c("Working Age", "Pension Age", "Unemployed/Young")
data_clean$income_group <- factor(data_clean$cluster,
labels = cluster_labels[order(cluster_summary$mean_income)])
ggplot(data_clean, aes(x = age, y = income, color = income_group)) +
geom_point(alpha = 0.5) +
scale_color_viridis_d() +
labs(title = "Age vs Income by Cluster",
x = "Age",
y = "Income (HUF)",
color = "Income Group") +
custom_theme
Descriptive Statistics
summary_stats <- summary(data)
kable(summary_stats, caption = "Summary Statistics of the Dataset (Outliers Removed)") %>%
kable_styling(bootstrap_options = c("striped", "hover", "condensed"), full_width = FALSE)| age | city | occupation | gender | income | starting_age | retirement_age | age_group | cluster | income_group | |
|---|---|---|---|---|---|---|---|---|---|---|
| Min. :14.00 | Length:7375 | Length:7375 | Length:7375 | Min. :379808 | Min. :14.00 | Min. :60.00 | 30 : 397 | Min. :1 | Working Age :7375 | |
| 1st Qu.:29.00 | Class :character | Class :character | Class :character | 1st Qu.:529234 | 1st Qu.:18.00 | 1st Qu.:65.00 | 32 : 377 | 1st Qu.:1 | Pension Age : 0 | |
| Median :39.00 | Mode :character | Mode :character | Mode :character | Median :579982 | Median :19.00 | Median :67.00 | 34 : 377 | Median :1 | Unemployed/Young: 0 | |
| Mean :40.14 | NA | NA | NA | Mean :582750 | Mean :18.89 | Mean :67.07 | 28 : 369 | Mean :1 | NA | |
| 3rd Qu.:50.00 | NA | NA | NA | 3rd Qu.:632598 | 3rd Qu.:20.00 | 3rd Qu.:69.00 | 42 : 365 | 3rd Qu.:1 | NA | |
| Max. :73.00 | NA | NA | NA | Max. :864318 | Max. :24.00 | Max. :75.00 | 40 : 360 | Max. :1 | NA | |
| NA | NA | NA | NA | NA | NA | NA | (Other):5130 | NA | NA |
ggplot(data %>% filter(age >= 18), aes(x = income, fill = gender)) +
geom_density(alpha = 0.6) +
scale_fill_viridis_d() +
labs(title = "Income Distribution by Gender",
subtitle = "(working age only)",
x = "Income (HUF)",
y = "Density") +
custom_theme
ggplot(data, aes(x = age, y = income, color = gender)) +
geom_point(alpha = 0.1, width = 0.2) +
scale_color_viridis_d() +
labs(title = "Income Distribution by Age",
subtitle = "(working age only)",
x = "Age",
y = "Income (HUF)") +
custom_theme
ggplot(data %>% filter(age >= 18), aes(x = reorder(occupation, income, FUN = median), y = income, color = occupation)) +
geom_boxplot(alpha = 0.7) +
geom_jitter(alpha = 0.1, width = 0.2) +
scale_color_viridis_d() +
coord_flip() +
labs(title = "Income Distribution by Occupation",
subtitle = "(working age only)",
x = "Occupation",
y = "Income (HUF)") +
custom_theme
ggplot(data %>% filter(age >= 18), aes(x = reorder(city, income, FUN = median), y = income, fill = city)) +
geom_violin(alpha = 0.7) +
geom_boxplot(width = 0.2, alpha = 0.5) +
scale_fill_viridis_d() +
coord_flip() +
labs(title = "Income Distribution by City",
subtitle = "(working age only)",
x = "City",
y = "Income (HUF)") +
custom_theme
ggplot(data %>% filter(age >= 18), aes(x = age, y = income, color = gender)) +
geom_point(alpha = 0.3) +
geom_smooth(method = "loess", se = TRUE) +
scale_color_viridis_d() +
labs(title = "Relationship between Age and Income",
subtitle = "(working age only)",
x = "Age",
y = "Income (HUF)") +
custom_theme
income_by_category <- data %>%
filter(age >= 18) %>%
group_by(occupation, city, gender) %>%
summarise(
mean_income = mean(income),
count = n(),
.groups = 'drop'
) %>%
arrange(desc(mean_income))
# heatmap
ggplot(income_by_category, aes(x = city, y = occupation, fill = mean_income)) +
geom_tile() +
scale_fill_viridis(name = "Mean Income (HUF)") +
facet_wrap(~gender) +
labs(title = "Mean Income by Occupation, City, and Gender",
subtitle = "(working age only)",
x = "City",
y = "Occupation") +
custom_theme +
theme(axis.text.x = element_text(angle = 45, hjust = 1),
strip.text = element_text(face = "bold"))
top_earners <- income_by_category %>%
arrange(desc(mean_income)) %>%
head(10)
kable(top_earners,
caption = "Top 10 Highest Earning Combinations",
digits = 0) %>%
kable_styling(bootstrap_options = c("striped", "hover", "condensed"),
full_width = FALSE)| occupation | city | gender | mean_income | count |
|---|---|---|---|---|
| Software Developer | Budapest | Male | 745832 | 99 |
| Software Developer | Budapest | Female | 722333 | 100 |
| Doctor | Budapest | Male | 720956 | 61 |
| Doctor | Budapest | Female | 696091 | 77 |
| Software Developer | Debrecen | Male | 695662 | 43 |
| Software Developer | Szeged | Male | 694999 | 30 |
| Manager | Budapest | Male | 692519 | 107 |
| Software Developer | Szeged | Female | 684845 | 43 |
| Engineer | Budapest | Male | 675140 | 133 |
| Software Developer | Debrecen | Female | 674727 | 40 |
Hypothesis Testing
Parametric Tests
1. Gender Income Difference (t-test)
# Test if there's a significant difference in income between genders
t_test_result <- t.test(income ~ gender, data = data)
print(t_test_result)##
## Welch Two Sample t-test
##
## data: income by gender
## t = -10.655, df = 7367.6, p-value < 2.2e-16
## alternative hypothesis: true difference in means between group Female and group Male is not equal to 0
## 95 percent confidence interval:
## -21472.33 -14798.96
## sample estimates:
## mean in group Female mean in group Male
## 573558.0 591693.72. City Income Differences (ANOVA)
# Test if there are significant differences in income between cities
city_anova <- aov(income ~ city, data = data)
print(summary(city_anova))## Df Sum Sq Mean Sq F value Pr(>F)
## city 7 1.356e+13 1.937e+12 539.9 <2e-16 ***
## Residuals 7367 2.643e+13 3.587e+09
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 13. Occupation Income Differences (ANOVA)
# Test if there are significant differences in income between occupations
occupation_anova <- aov(income ~ occupation, data = data)
print(summary(occupation_anova))## Df Sum Sq Mean Sq F value Pr(>F)
## occupation 9 1.545e+13 1.717e+12 515.5 <2e-16 ***
## Residuals 7365 2.453e+13 3.331e+09
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Non-parametric Tests
1. Gender Income Distribution (Kolmogorov-Smirnov Test)
# Test if income distributions differ between genders
male_income <- data$income[data$gender == "Male"]
female_income <- data$income[data$gender == "Female"]
ks_test <- ks.test(male_income, female_income)
print(ks_test)##
## Asymptotic two-sample Kolmogorov-Smirnov test
##
## data: male_income and female_income
## D = 0.10969, p-value < 2.2e-16
## alternative hypothesis: two-sided2. Income Distribution by City (Kruskal-Wallis Test)
# Test if income distributions differ between cities
kruskal_test <- kruskal.test(income ~ city, data = data)
print(kruskal_test)##
## Kruskal-Wallis rank sum test
##
## data: income by city
## Kruskal-Wallis chi-squared = 2574.8, df = 7, p-value < 2.2e-16Regression Analysis
Multiple Linear Regression
# Convert categorical variables to factors
data_working_age <- data %>% filter(age >= 18)
data_working_age$city <- as.factor(data_working_age$city)
data_working_age$occupation <- as.factor(data_working_age$occupation)
data_working_age$gender <- as.factor(data_working_age$gender)
# Fit the model
model1 <- lm(income ~ age + I(age^2) + city + occupation + gender, data = data_working_age)
# Create a beautiful model summary table
model_summary <- summary(model1)
kable(tidy(model_summary), caption = "Multiple Linear Regression Results (Working Age Only)") %>%
kable_styling(bootstrap_options = c("striped", "hover", "condensed"), full_width = FALSE)| term | estimate | std.error | statistic | p.value |
|---|---|---|---|---|
| (Intercept) | 434779.5680 | 3042.221285 | 142.91517 | 0 |
| age | 8492.2241 | 146.411115 | 58.00259 | 0 |
| I(age^2) | -79.2278 | 1.733131 | -45.71369 | 0 |
| cityDebrecen | -50118.9464 | 885.873428 | -56.57574 | 0 |
| cityEger | -100673.3825 | 1223.767096 | -82.26515 | 0 |
| cityGyőr | -98817.6008 | 1136.347121 | -86.96075 | 0 |
| cityMiskolc | -99847.5434 | 1064.213618 | -93.82284 | 0 |
| cityPécs | -98754.3471 | 1121.335644 | -88.06850 | 0 |
| citySzeged | -49284.8644 | 997.322532 | -49.41718 | 0 |
| citySzombathely | -99991.5016 | 1348.646293 | -74.14212 | 0 |
| occupationChef | -42406.4096 | 1535.934166 | -27.60952 | 0 |
| occupationDoctor | 59742.5726 | 1554.207886 | 38.43924 | 0 |
| occupationDriver | -50964.5811 | 1535.962289 | -33.18088 | 0 |
| occupationEngineer | 18051.4651 | 1251.405236 | 14.42496 | 0 |
| occupationManager | 37826.6478 | 1351.616466 | 27.98623 | 0 |
| occupationNurse | -32257.7329 | 1187.823213 | -27.15702 | 0 |
| occupationSales Representative | -60650.2152 | 1057.574345 | -57.34842 | 0 |
| occupationSoftware Developer | 88380.8010 | 1335.055549 | 66.20009 | 0 |
| occupationTeacher | -22511.0613 | 1128.935083 | -19.94008 | 0 |
| genderMale | 18870.6304 | 582.757427 | 32.38162 | 0 |
# Enhanced model diagnostics
par(mfrow = c(2,2))
plot(model1, col = income_palette[1], pch = 19, cex = 0.7)
Polynomial Regression for Age-Income Relationship
# Fit polynomial regression
model2 <- lm(income ~ poly(age, 3), data = data_working_age)
# Create a static plot
ggplot(data_working_age, aes(x = age, y = income)) +
geom_point(alpha = 0.1, color = income_palette[1]) +
geom_smooth(method = "lm", formula = y ~ poly(x, 3),
color = income_palette[5], fill = income_palette[5], alpha = 0.2) +
labs(title = "Polynomial Regression: Age vs Income",
subtitle = "Cubic polynomial fit with confidence interval (Working Age Only)",
x = "Age",
y = "Income (HUF)") +
custom_theme
# Model comparison
model_comparison <- data.frame(
Model = c("Multiple Linear", "Polynomial"),
R_squared = c(summary(model1)$r.squared, summary(model2)$r.squared),
Adj_R_squared = c(summary(model1)$adj.r.squared, summary(model2)$adj.r.squared)
)
kable(model_comparison, caption = "Model Comparison (Working Age Only)") %>%
kable_styling(bootstrap_options = c("striped", "hover", "condensed"), full_width = FALSE)| Model | R_squared | Adj_R_squared |
|---|---|---|
| Multiple Linear | 0.8851681 | 0.8848691 |
| Polynomial | 0.1541084 | 0.1537614 |
Predictions
# Create a sample prediction
new_data <- data.frame(
age = 35,
city = "Budapest",
occupation = "Software Developer",
gender = "Male"
)
# Predict income
prediction <- predict(model1, newdata = new_data, interval = "prediction")
print(prediction)## fit lwr upr
## 1 742204.8 693333.9 791075.6Summary and Conclusions
The analysis of the simulated Hungarian income data reveals several interesting patterns:
- There is a significant gender pay gap, with males earning more on average than females.
- Income varies significantly across different cities, with Budapest showing the highest average income.
- Occupation has a strong effect on income, with software developers and doctors earning the most.
- Age shows a non-linear relationship with income, peaking in the middle age range.
- The regression models explain a significant portion of the income variation.
The analysis demonstrates the complex interplay between various factors affecting income levels in Hungary. The findings suggest that both demographic factors (age, gender) and professional factors (occupation, location) play important roles in determining income levels.